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Applications of Machine Learning in Material Science & Chemical Engineering

Chemical reactions and phase transformations underpin phenomena ranging from cosmological processes, to the emergence of life on Earth, to modern technologies and are therefore of tremendous interest for both basic and applied sciences.

Here I provide a review of recent advancements in the intersection between computational mathematics, material science & chemical engineering.



This study made a multitude of notable contributions:

1. Automated Data Labeling

Fast Fourier Transforms were used to label the data in an automated & reliable manner.

2. Defect Identification

Deep learning was used to map breeches in lattice periodicity to atomic defects in Mo-doped WS2 — effectively identifying atomic defects in the data.

3. Defect Classification

A Gaussian Mixture Model was then used to probabilistically cluster defects. On examination, these clusters map well to theoretical interpretations.

4. Contrast Learnt Abstractions with Theoretical Physics

Using the above as data processing, researchers were then able to:

  1. Identify & describe spatiotemporal behaviour in recurring defect types.
  2. Scrutinize diffusion of select (S vacancy) defects.
  3. Offer detailed study of the transition dynamics in Mo-S complexes.

Through this application, it is demonstrated that end-to-end machine learning can be used to discover patterns that are well understood through the lens of conventional physics/chemistry. One possible ambitious next step of this work be to encode various learning problems with tranistion dynamics or phase evolution characteristics with laws understood in the physics literature — with the aim of constraining the search space to answer questions outside of the scope of our current understanding.

Deep learning analysis of defect and phase evolution during electron beam-induced transformations in WS2

Reference paper

Modern advancements in scanning transmission electron microscopy (STEM) are enabling direct visualization of dynamic phenomena at the atomic level, some techniques are even offering methods of controlled movement of the atoms through a lattice.

These imaging & transformation techniques have the potential to contribute to an eclectic range of problems (from medical diagnosis to aerospace engineering, to the smart fuels to tackle energy crisis) — material science may, in this way, play an important role in many of the scientific, engineering & medical challenges of our generation.

Actualizing this potential is not without engineering challenges. One of which (of increasing importance) is the autonomous processes of these rich data sources.

In this paper, the authors:

“Analyze the phase evolution of Mo-doped WS2 during electron beam irradiation.”

With the goals of:

“Developing a deep-learning network for rapid analysis of this dynamic data, analyze transformation pathways, create a library of defects, and explore minute distortions in local atomic environment around the defects of interest, ultimately building a complete framework for exploring point-defect dynamics and reactions.”

Raw data: STEM Imaging

STEM (scanning transmission electron microscopy) “movies” were taken of Mo-doped WS2 during electron beam irradiation, as depicted here:

Figure 1. STEM data of e-beam irradiation in Mo-doped WS2. a. provides a Ball-and-stick representation of WS2. b-to-e capture 4 frames of a STEM movie in the data: showing the defects & lattice transformation over time.

Existing methods of analyzing material defects are manual & often inefficient.

Statistical Enquiry

The modeling process consisted of various distinct steps, each with accompanying techniques & analysis.

Convolutional Neural Network

Defect Identification

In order to develop autonomous tools, manual processes need to be superseded. Is it known that each defect type is associated with a periodic distortion in the lattice structure, thus a single frame from the STEM movie can be used as input data to predict each defect species. A CNN (ConvNet) is used to map the input frames to defects. It was also shown that the network could detect defects that are omitted from the training set.


The CNN model was trained to map the input data to an output space (image) of equal size, given the probability of each pixel in the image belonging to a certain defect class.

The network is able to:

  1. Localize atomic-scale lattice disorders.
  2. Return the location of defects.
  3. Generalize to unseen defect structures.

Fast Fourier Transforms

Image Preprocessing

FFT (Fast Fourier Transforms) — and inverse FFT — were used to label the data for the CNN. The FFT can reliably isolate regions of defects, providing pixel labeling used to train the function approximator.

a. Aframe from the STEM movie (input space). b. An illustration of the FFT applied to a frame. c. The labeled data after FFT processing. d. A description of the CNN architecture.

Once trained, the CNN boasts a number of advantages over the FFT:

  • CNN is far more efficient at runtime.
  • CNN is more robust — rotation & scale-invariant.
  • CNN generalizes better to unknown defect types (lattice distortions).

Gaussian Mixture Model (GMM)

Defect Classification

Now that the raw video data can be reliably autonomously labeled by the FFT → CNN batch updating. Atomic defects can be identified in dynamic STEM data on Mo-doped WS2. Although this worked with great accuracy, the analysis was taken further to cluster the defects — & thereafter assess whether this fits our current theoretical models.

A GMM was used to probabilistically cluster the (unlabelled) defects into 5 classes.

This classification allows us to begin to map the learnt representations to well-understood theoretical models from the chemistry/physics literature. It can be observed that:

  • Class 1 and 3 correspond to a substitutional atom in W sublattice with a lower Z number, which we interpret as Mo dopant (MOw).
  • Classes 4 and 5 are associated with a vacancy in the W sublattice (Vw) and in the S sublattice (Vs), respectively.

Variation & Diffusion Coefficient Estimation

Analysising Individual Defects

Further, it is possible to estimate diffusion characteristics of the selected defect species using the classification output of the GMM. The diffusion properties of S vacancies are examined.

Specific classes of defects were then projected from 3-dimensional spatiotemporal diagram to a 2-dimensional representation by select “windows” of the specific classes of defects (as show in figure 4).

Density-based cluster was then applied to estimate the variance of each distribution & a diffusion coefficient within a framework of a random walk model in 2-dimensions.

Principal Components Analysis (PCA)

Local Crystallography Analysis

The analysis was taken further, conducting local crystallography analysis on classes 1 & 3. This was done by:

  1. Mo dopant defects where (GMM class 1 & 3 extractions) were selected.
  2. A deep-learning-based “atom finder” was employed to extract the atom positions in thousands of noisy images.
  3. These atom configurations were then aggregated to produced an image of the average defect configuration. The averaged images provide the central Mo atom and six W neighbor atoms for each defect class.
  4. PCA was then applied to exact the first 2 eigenmodes of the averaged images — extracting vectors of maximum variation.

It appears plausible that there is significant variation in the relative position of the central Mo atom with respect to neighbouring W atoms. This may be attributed to the presence of S vacancies next to Mo dopant.

Markov Process

Transition Dynamics

The PCA analysis & lattice symmetry was then used to split the 2 classes (class 1 and 3) into 4 subclasses. These subclasses represent:

  • MOw: undistorted
  • (MOw + Vs): 3 MOw + Vs complexes.

Defect State-Transition Analysis

As represented in figure 6a, each “flow” describes a defect moving between states. As a consequence we can analyze the transition between states as a Markov process.

Figure 6.c and 6.d provide the transition dynamic schemata & matrix respectively (describing the probability of moving between defect states).

From a physics/chemistry theoretical perspective, it can be argued that transitions between states correspond to the lower diffusion barrier of an S vacancy (single atom vacancy). It appears that these chemical structures are short-lived & unstable (which may in-itself be a notable contribution).

Finally, the data suggests that variation in supply of S vacancies from different lattice directions can explain defect state transition probabilities.


This study detailed an end-to-end ML approach to modelling phase evolution & defects during electron beam-induced transformations in WS2, by:

1. Automated Data Labeling (FFTs)

2. Defect Identification (CNN)

3. Defect Classification (GMM)

4. Contrast Learnt Abstractions with Theoretical Physics (PCA & Markov Model Analysis)

Reference paper available here.



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Zach Wolpe

Zach Wolpe

Statistician, scientist, technologist — writing about stats, data science, math, philosophy, poetry & any other flavours that occupy my mind. Get in touch